Reduced-order filter for stochastic bilinear systems with multiplicative noise

(1)

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Reduced-order filter for stochastic bilinear systems with multiplicative noise

Souheil Halabi, Harouna Souley Ali, Hugues Rafaralahy, Michel Zasadzinski, Mohamed Darouach

To cite this version:

Souheil Halabi, Harouna Souley Ali, Hugues Rafaralahy, Michel Zasadzinski, Mohamed Darouach.

Reduced-order filter for stochastic bilinear systems with multiplicative noise. 5th IFAC Symposium

on Robust Control Design, ROCOND 2006, Jul 2006, Toulouse, France. pp.CDROM. �hal-00098058�

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Reduced-order filter for stochastic bilinear systems with multiplicative noise

S. Halabi, H. Souley Ali, H. Rafaralahy, M. Zasadzinski, M. Darouach Universit´e Henri Poincar´e − Nancy I,

CRAN UMR 7039 − CNRS IUT de Longwy,

186, rue de Lorraine, 54400 Cosnes et Romain, FRANCE e-mail : { shalabi,mzasad } @iut-longwy.uhp-nancy.fr

Abstract— This paper deals with the design of a reduced- orderH∞filter for a stochastic bilinear systems with a pre- scribedH∞norm criterion. The problem is transformed into the search of a unique gain matrix by using a Sylvester-like condition on the drift term. The considered system is bilinear in control and with multiplicative noise in the state and in the measurement equations. The approach is based on the resolution of LMI and is then easily implementable.

Keywords— Reduced-orderH∞filter, Itˆo’s formula, Stochas- tic systems, Bilinear systems, Lyapunov function.

I. Introduction

The bilinear systems represent sometimes a good mean to physical systems modeling when the linear representa- tion is not sufficiently significant. The stochastic systems get a great importance during the last decades as shown by numerous references (Kozin, 1969; Has’minskii, 1980;

Florchinger, 1995; Mao, 1997; Carravettaet al., 2000; Ger- maniet al., 2002; Xu and Chen, 2003).

Generally, bilinear stochastic system designs a stochastic system with multiplicative noise instead of additive one (Carravetta et al., 2000; Germani et al., 2002). The full and the reduced-order_H_∞ filtering for stochastic systems with multiplicative noise has been treated in many papers (Hinrichsen and Pritchard, 1998; Gershonet al., 2001; Xu and Chen, 2002; Stoica, 2002). Notice that the measurement equation in (Xu and Chen, 2002; Stoica, 2002) is not corrupted by noise. The problem is solved in terms of two LMIs and a coupling non convex rank constraint.

In this paper the problem of reduced-order_H_∞filtering for a larger class of stochastic systems than those studied in the papers cited above is considered since the studied systems are with multiplicative noise and multiplicative control input (the bilinearity is also between the state and the control input). The measurements are subjected to a multiplicative noise too. Notice that, as in the deterministic case, the multiplicative control input affects the observability of the system.

The purpose is to design a reduced-order_H_∞filter for such a system. We first use a “unbiasedness” (decoupling) condition on the drift part of the estimation error and a change of variable on the control input. Then applying the Itˆo formula and LMI method permit to reduce the problem to the search of a unique gain matrix. The reduced-order stochastic filter matrices are then computed using this gain.

Throughout the paper, E represents expectation operator with respect to some probability measure_P. _!X, Y"=X^TY represents the inner product of the vectors X, Y ∈ IRⁿ.

herm(A) stands forA+A^T. L2`

Ω,IR^k´is the space of square-integrableIR^k-valued func- tions on the probability space(Ω,F,P)whereΩis the sample space,_F is a σ-algebra of subsets of the sample space called events and_Pis the probability measure on_F. (F^t)_t!0 denote an increasing family of σ-algebras (F^t) ∈ F. We also denote byLb2`

[0,∞) ; IR^k´the space of non-anticipatory square-integrable stochastic process f(.) = (f(t))_t_∈_[0,_∞₎ in IR^k with respect to(F^t)_t_∈_[0,_∞₎ satisfying

%f%²Lb2=E

Z _∞

0 %f(t)%²dt ff

<∞ where_%.%is the well-known Euclidean norm.

II. Problem statement

Let us consider the following stochastic bilinear system 8>

><

>>

:

dx(t) = (At0x(t) +u1(t)At1x(t)) dt +B0v(t) dt+Aw0x(t) dw0(t) dy(t) = Cx(t) dt+J1x(t) dw1(t)

z(t) = Lx(t)

(1)

where x(t) ∈IRⁿ is the state vector, y(t) ∈IR^p is the out- put, u1(t) ∈ IR is the control input, z(t) ∈ IR^r is a linear combination of the state vector with r < nand v(t)∈IR^q is the perturbation signal. Without loss of generalityL is assumed to be a full row rank matrix. wi(t) is a Wiener process verifying (Has’minskii, 1980)

E (dwi(t)) = 0,E(dwi(t)²) = dt, fori= 0,1, (2a) E (dw0(t) dw1(t)) =E(dw1(t) dw0(t)) =ϕdt,

with_|ϕ|<1. (2b) As in the most cases for physical processes, we assume that the stochastic bilinear system (1) has known bounded control input, i.e. u1(t)∈Γ⊂IR, where

Γ ={u1(t)∈IR | u1 min!u1(t)!u1 max}. (3) The study made here can be easily generalized for the case where there aremcontrol inputs.

First, we introduce the following definition and assumption.

Definition 1. (Kozin, 1969; Has’minskii, 1980) The stochastic system (1) with v(t) ≡ 0 is said to be asymp- totically mean-square stable if all initial statesx(0)yields

tlim→∞E%x(t)%²= 0, ∀u1(t)∈Γ. (4)

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Assumption 1. The stochastic bilinear system (1) is as- sumed to be asymptotically mean-square stable.

In this paper, the aim is to design a reduced-order filter in the following form

dη(t) = (M0+u1(t)M1)η(t) dt

+ (N0+u1(t)N1) dy(t) (5) where η(t)∈IR^r is the filter state withr < n and the ma- tricesMi andNi (fori= 0,1) are to be determined.

Then the following problem is considered.

Problem 1. Given a real γ > 0, the goal is to design a asymptotically mean-square stable reduced-order _H_∞ filter (5) such that the augmented state[x^T(t) e^T(t)]^T is asymp- totically mean-square stable and the following _H_∞ perfor- mance

%e(t)%²Lb2!γ%v(t)%²Lb2 (6) is achieved from the disturbance v(t) to the filtering error e(t) =z(t)−η(t).

Let us consider the following estimation error

e(t) =Lx(t)−η(t). (7) So the estimation error dynamics becomes

de(t) = (M0+M1u1(t))e(t) dt+LB0v(t) dt +{(LAt0−M0L−N0C)

+ (LAt1−M1L−N1C)u1(t)}x(t) dt +LAw0x(t) dw0(t)

−((N0+u1(t)N1)J1x(t) dw1(t). (8) In order to supress the direct effect of the statex(t)on the drift part of the filtering error, we consider the following Sylvester-like conditions

LAti−MiL−NiC= 0, i= 0,1. (9) Let us consider the following augmented state vector

ξ^T(t) =ˆ

x^T(t) e^T(t)˜

. (10)

Then under (9), the dynamics of the augmented system is given by

dξ(t) = (A^t0+A^t1u1(t))ξ(t) dt+B⁰v(t) dt +A^w0ξ(t) dw0(t)

+ (A^w1+A^w2u1(t))ξ(t) dw1, (11) with

A^ti=

»Ati 0 0 Mi

–

, for i= 0,1, B0=

»B0

LB0

–

, Aw0=

»Aw0 0 LAw0 0 –

, A^w1=

» 0 0

−N0J1 0 –

, A^w2=

» 0 0

−N1J1 0 –

. In the sequel the relations (9) are used to express the filter matrices through a single gain matrix.

In fact, sinceL is a full row rank matrix, relations (9) are equivalent to

(LAti−MiL−NiC)ˆ

L^† (In−L^†L)˜

= 0,

for i= 0,1. (13) whereL^†is a generalized inverse of matrixLsatisfyingL= LL^†L(Lancaster and Tismenetsky, 1985) (sincerankL=r, we haveLL^†=Ir).

Relations (13) give

0 =LAtiL^†−Mi−NiCL^† for i= 0,1, (14a) 0 =LAi−NiC for i= 0,1, (14b) where

Ai=Ati(In−L^†L) for i= 0,1, (15a)

C=C(In−L^†L). (15b)

The relation (14a) gives

Mi=Ai−NiC, for i= 0,1, (16) where

Ai =LAtiL^†, for i= 0,1, (17a)

C =CL^†. (17b)

The relation (14b) becomes

KΣ =LA, (18)

where

K=ˆ N0 N1˜

, (19)

A=ˆ A0 A1˜

, (20)

Σ =

»C 0 0 C –

, (21)

and a general solution to equation (18), if it exists, is given by

K=LAΣ^†+Z(I2p−Σ Σ^†), (22) where

Z=ˆ Z0 Z1˜

, (23)

is an arbitrary matrix of appropriate dimensions.

III. Tranformation of the bilinear system filtering problem into an uncertain one As in (Zasadzinskiet al., 2003), let us introduce a change of variable on the controlu1(t) as follows

u1(t) =α1+σ1ε1(t) (24) whereα1∈IRand σ1∈IRare given by

α1= 1

2(u1min+u1max),σ1=1

2(u1max−u1min). (25) The new “uncertain” variable is ε1(t)∈ Γ⊂IR where the polytopeΓis defined by

Γ ={ε1(t)∈IR | ε1min=−1!ε1(t)!ε1max= 1}. (26)

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Then the error dynamics (8) can be rewritten as de(t) =“

A^t−ZC^t+ (eA^t−ZCe^t)∆ε(ε1(t))He

”e(t) dt +B⁰v(t) dt+A^w0x(t) dw0(t) +`

Aw(11)−ZAw(12)

+(eA^w(11)−ZAe^w(12))∆x(ε1(t))Hx

”x(t) dw1(t) (27) where

At=A0+α1A1−LAΣ^†Λ, Ct= (I2p−Σ Σ^†)Λ, Ae^t=σ1A1−LAΣ^†Λ, Ce^t= (I2p−Σ Σ^†)Λ, B⁰¹=LB0, A^w0=LAw0, Aw(11)=LAΣ^†Ψα, Aw(12)= (I2p−Σ Σ^†)Ψα, Aew(11)=LAΣ^†Ψσ, Aew(12)= (I2p−Σ Σ^†)Ψσ, and

Λ =

» CL^† α1CL^†

–

, Ψα=

»−J1

−αJ1

–

, Ψσ=

» 0

−σJ1

– , He=Ir, Hx=In,

∆ε(ε1(t)) =ε1(t)Ir, ∆x(ε1(t)) =ε1(t)In. Using the definition (26), the matrix∆ε(ε1(t))and∆x(ε1(t)) satisfy

%∆ε1(ε1(t))%!1, and _%∆x(ε1(t))%!1. (28) Using (24), the system state equation (see (1)) becomes

dx(t) = (At0+α1At1+σ1ε1(t)At1)x(t) dt

+B0v(t) dt+Aw0x(t) dw0(t). (29) So the augmented system (11) is rewritten as

dξ(t) =“

Abt0+ ∆Abt0(t)”

ξ(t) dt+Bb0v(t) dt +Abw0ξ(t) dw0(t)

+“

Abw1+ ∆Abw1(t)”

ξ(t) dw1(t) (30) where

Abt0=

»At0+α1At1 0 0 A^t−ZC^t

– ,

∆Abt0(t) =H1∆ξ(ε1(t))Ht, Bb0=

»B0

B⁰¹ –

, Abw0=

»Aw0 0 A^w0 0 –

, Abw1=

» 0 0

A^w(11)−ZA^w(12) 0 –

,

∆Abw1(t) =H2∆ξ(ε1(t))Hw, H1=

"

σ1At1 0 0 Ae^t−ZCe^t

# , H2=

"

0 Aew(11)−ZAew(12)

# ,

Ht=

»In 0 0 Ir

–

, Hw=ˆ In 0˜

,∆ξ(ε1(t)) =ε1(t).

Notice that from (26),∆ξ(ε1(t))satisfy

%∆ξ(ε1(t))%!1. (31)

IV. Synthesis of the reduced-order filter Consider the following system obtained from (30)

8>

><

>>

:

dξ(t) = “ b

At0+∆Abt0(t)”

ξ(t) dt+Bb0v(t) dt +Abw0ξ(t) dw0(t)+“

b

Aw1+∆Abw1(t)”

ξ(t) dw1(t) e(t) = Cξ(t)b

(32)

whereC^b=ˆ 0 Ir˜.

Then the following theorem is given for the filter synthesis.

Theorem 1. The reduced-order_H_∞ filtering problem 1 is solved for the system (1) with the filter (5) such that the augmented system (32) is asymptotically mean-square sta- ble and verifies the_H_∞ performance (6) if, for some reals µ1 >0, µ2 >0 andµ3 >0 there exist matrices P1 =P₁^T >

0∈IRⁿ^×ⁿ,P2=P₂^T >0∈IR^r^×^r,P3∈IRⁿ^×^r, G2∈IR^r^×^2pand G3∈IR^n×2p such that

2 66 66 66 66 66 4

(1,1) (1,2) P₁B₀+P₃B01 σ₁P₁A_t1 (1,2)^T (2,2) P₃^TB0+P2B01σ1P₃^TAt1 B^T₀P1+B^T01P₃^T B₀^TP3+B^T01P2 −γ²Iq 0

σ1A^T_t1P1 σ1A^T_t1P₃^T 0 −µ1In eA^TtP₃^T−eC^TtG^T₃ eA^TtP2−eC^TtG^T₂ 0 0

(1,6) 0 0 0

(1,7) 0 0 0

(1,8)^T 0 0 0

(1,9)^T 0 0 0

(1,10)^T 0 0 0

0 0 0 0

P3eAt−G3Cet (1,6) (1,7) (1,8) (1,9) (1,10) 0 P2eAt−G2Cet 0 0 0 0 0 0

0 0 0 0 0 0 0

−µ1Ir 0 0 0 0 0 0

0 −P₁ −P₃ 0 0 0 0

0 −P₃^T −P2 0 0 0 0

0 0 0 −µ₃In 0 0 0

0 0 0 0 −P1 −P3 (11,9)

0 0 0 0 −P3^T −P2 (11,10)

0 0 0 0 (11,9)^T (11,10)^T −µ2In

3 77 77 77 77 75

<0 (33)

where

(1,1) = (µ1+µ2+ϕµ3)In+ herm{P1Aα1

+ϕ“ A^T_w0`

P3Aw(11)−G3Aw(12)

´ +A^Tw0 `

P2Aw(11)−G2Aw(12)´´¯

, (1,2) =A^Tα1P3+P3A^t−G3C^t,

(2,2) = herm (P2A^t−G2C^t) + (1 +µ1)Ir, (1,6) =A^Tw0P1+A^Tw0P3^T,

(1,7) =A^Tw0P3+A^Tw0P2, (1,8) =ϕ¹²“

A^T_w0“

P3Aew(11)−G3Aew(12)

” +A^Tw0

“

P2Aew(11)−G2Aew(12)

””

, (1,9) =A^Tw(11)P₃^T−A^Tw(12)G^T₃,

(1,10) =A^Tw(11)P2−A^Tw(12)G^T₂, (11,9) =P3Aew(11)−G3Aew(12), (11,10) =P2Ae^w(11)−G2Ae^w(12),

Aα1=At0+α1At1,

and such that the gain matricesG2 andG3 are the solution of the following equation

»G2

G3

–

=

»P2

P3

–

Z. (34)

(5)

Proof. Consider the following Lyapunov function

V(ξ) =ξ^TPξ, (35)

where

P=

»P1 P3

P₃^T P2

–

. (36)

Applying Itˆo formula (Mao, 1997) to the system (30) (or (32)), we get

dV(ξ(t)) =LV(ξ(t)) dt+ 2ξ^T(t)PΨ(t)ξ(t) (37) where

Ψ(t) =Abw0dw0(t) +“ b

Aw1+ ∆Abw1(t)”

dw1(t), (38) and

LV(ξ(t)) dt=2ξ^T(t)““

Abt0+∆Abt0(t)”

ξ(t)+Bb0v(t)” dt

+ξ^T(t)!PΨ(t),Ψ(t)"ξ(t). (39) By replacing (38) and (39), the relation (37) becomes

dV(ξ(t)) = 2ξ^T(t)P““

Abt0+ ∆Abt0(t)”

ξ(t) +Bb0v(t)” dt +ξ^T(t)Ab^Tw0PAbw0ξ(t) dw0(t)²

+ξ^T(t)“

Abw1+ ∆Abw1(t)”T

P

×“

Abw1+ ∆Abw1(t)”

ξ(t) dw1(t)² +ξ^T(t)Ab^T_w0P“

Abw1+ ∆Abw1(t)”

ξ(t) dw0(t) dw1(t) +ξ^T(t)“

Abw1+ ∆Abw1(t)”T

PAbw0ξ(t) dw1(t) dw0(t) + 2ξ^T(t)PAbw0ξ(t) dw0(t)

+ 2ξ^T(t)P“ b

Aw1+ ∆Abw1(t)”

ξ(t) dw1(t). (40) Using the majoration lemma (Wanget al., 1992), it can be shown that

2ξ^T(t)P∆Abt0(t)ξ(t)

!ξ^T(t)“

µ⁻₁¹PH1H₁^TP+µ1H_t^THt

”ξ(t), (41)

“Abw1+∆Abw1(t)”T

P“

Abw1+∆Abw1(t)”

!Ab^T_w1“

P⁻¹−µ⁻₂¹H2H₂^T”−1

Abw1+µ2H_w^THw, (42) 2ξ^T(t)Ab^T_w0P∆Abw1(t)ξ(t)

!ξ^T(t)“

µ⁻¹₃ Ab^Tw0PH2H2^TPAbw0+µ3Hw^THw

”

ξ(t). (43) Now, taking the expectation of (40) (see (Mao, 1997)) and using the relations (2) and the last three inequalities, then E{dV(ξ(t))}can be bounded as

E{dV(ξ(t))}!E

ˆ

ξ(t)^T v(t)^T˜ Θ1

»ξ(t) v(t) –

dt

ff (44)

where Θ1 =

"

PAbt0+Ab_t0^TP+C^TC PBb0

Bb0^TP −γ²Iq

#

+µ1

»H_t^THt 0

0 0

– +µ⁻¹₁

»PH₁^TH1P 0

0 0

–

+

"

Ab^Tw1`

P⁻¹−µ⁻₂¹H2H2^T´−1Abw1 0

0 0

# +µ2

»H_w^THw 0

0 0

–

+ϕµ3

»H_w^THw 0

0 0

– +

"

Ab^Tw0PAbw0 0

0 0

#

+

"

ϕµ⁻¹₃ “ b

A^T_w0PH2H₂^TPAbw0

” 0

0 0

#

+ϕ

"

Ab^T_w0PAbw1+Ab^T_w1PAbw0 0

0 0

# . (45) Now, applying the Schur lemma (Boydet al., 1994),Θ1can be rewritten as

2 66 66 66 66 66 4

(1,1) PBb0 PH1 Ab^T_w0P Bb₀^TP −γ²Iq 0 0 H1^TP 0 −µ1In+r 0 PAbw0 0 0 −P ϕ¹²H₂^TPAbw0 0 0 0

PAbw1 0 0 0

0 0 0 0

ϕ¹²Ab^T_w0PH2 Ab^T_w1P 0

0 0 0

−µ3In+r 0 0

0 −P PH2

0 H₂^TP −µ2In

3 77 77 77 77 75

(46)

where

(1,1) =PAbt0+Ab^T_t0P+Cb^TCb+ (µ2+ϕµ3)H_w^THw

+µ1Ht^THt+ϕ“

Ab^Tw0PAbw1+Ab^Tw1PAbw0

” . (47) Once the LMI (33) is verified, the asymptotic mean-square stability of the system (32), for v(t) ≡ 0, can be proved using Schur lemma and the same method of (Souley Aliet al., 2005).

Now consider the following performance index Jξv=

Z_∞

0

E

“ξ^T(t)Cb^TCξ(t)b −γ²v^T(t)v(t)”

dt. (48)

WrittingJξv as Jξv=

Z _∞

0

n E

““

ξ^T(t)Cb^TCξ(t)b −γ²v^T(t)v(t)” dt

+ dV(ξ(t)))} −E(V(ξ(t))_t=_∞+E(V(ξ(t))_t=0. (49) Or, since E(V(ξ(t))_t=0 = 0 because ξ(0) = 0 and E(V(ξ(t))_t=_∞"0, this implies

Jξv!Z _∞

0

n E““

ξ^T(t)Cb^TCξ(t)b −γ²v^T(t)v(t)” dt

+ dV(ξ(t)))}. (50)

(6)

Now if the LMI (33) holds, then applying Schur lemma yields _"

Θ PBb0

Bb^T₀P 0

#

| {z }

Π

+

"

Cb^TCb 0 0 −γ²Iq

#

<0 (51)

with

Θ =PAbt0+Ab^Tt0P+µ1Ht^THt+ (µ2+ϕµ3)Hw^THw

+ϕ(Ab^T_w0PAbw1+Ab^T_w1PAbw0) +µ⁻₁¹PH₁^TH1P+ Ab^T_w0PAbw0+ϕµ⁻¹₃ “

Ab^T_w0PH2H₂^TPAbw0

”.

Therefore Jξv!Z _∞

0

E

„ˆ

ξ(t)^T v(t)^T˜ Π

»ξ(t) v(t) –

dt

+ˆ

ξ(t)^T v(t)^T˜"

Cb^TCb 0 0 −γ²Iq

# »ξ(t) v(t) –

dt

!

<0, so, if the LMI (33) holds the asymptotic mean-square stability and the_H_∞performance are proved. ❏

V. Numerical example

Consider the stochastic bilinear system (1) and suppose that the matrices have the following numerical value

At0= 2

4−1.5 1 −1 0.5 −2.5 1

0 −0.6 −3.5 3

5, B0 =

2

4−0.1 0.3

−1 0.2 0.6 0.5 3 5,

At1= 2

4−0.01 0.1 0 0 −0.05 0 0.15 0 −0.02

3 5,

Aw0= 2 4

1 0 0.2 0.5 0.3 −0.1

−0.2 0 0.2 3 5,

C=

»1 0 0 0 1 0 –

, L=

»0 −1 1 1 0 −1

– , J1=

»−0.03 0 −0.03 0 −0.01 0

– .

The controlu1(t)is defined as in (3), with u1 min=−5!u1(t)!u1 max= 6, and the initial stateξ(0) = [x^T(0) e^T(0) ]^T is

ξ(0) =ˆ

−1 0.5 1 0.5 −2˜T

.

The gain Z is obtaind for γ = 22 and µ1 = 7.4628, µ2 = 0.0057andµ3= 0.2476and is then given by

Z=

»−6179.565−6191.2468 47.380 47.2190

−1810.724 −1819.847 20.760 20.796 –

. Finally, the matrices of the reduced-order filter (5) are

M0=

»−9.291 −4.791

−4.862 −7.362 –

, M1=

»−0.041−0.020

−0.114−0.146 –

, N0=

»4.291 −7.391 5.862 −3.262 –

, N1=

» 0.171 0.009

−0.002−0.002 –

.

The following figures show the simulation results of the augmented system (32). The statex(t) and the estimation error e(t) are plotted. The disturbance signal v(t) is pre- sented with the error plots. The simulation is made for the controlu1(t) = 0.5 sin(3t) + 2, and the covariance factor between the Wiener processes defined in (2b)ϕ= 0.0215.

Time [sec]

Fig. 1. The actual statex(t).

v(t)

Time [sec]

Fig. 2. The errore(t) and the disturbancev(t).

VI. Conclusion

This paper provided a solution to the reduced-order _H∞

filtering problem for bilinear stochastic systems with multiplicative noise. The approach is based on a change of variable on the control input and on the using of a Sylvester- like condition on the drift term to transform the problem into a robust reduced-order stochastic filtering one. Using the LMI method and the Itˆo formula we reduced the problem to the search of a unique gain matrix. Then the filter matrices are computed through this gain.

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